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Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation

Published online by Cambridge University Press:  26 August 2021

Lihua Deng
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, People's Republic of China ([email protected], [email protected])
Haifeng Shang
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, People's Republic of China ([email protected], [email protected])

Abstract

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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